Theorem neii 2253

Description: Inference associated with df-ne 2252. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
neii.1	⊢ 𝐴 ≠ 𝐵

Assertion

Ref	Expression
neii	⊢ ¬ 𝐴 = 𝐵

Proof of Theorem neii

Step	Hyp	Ref	Expression
1		neii.1	. 2 ⊢ 𝐴 ≠ 𝐵
2		df-ne 2252	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbi 143	1 ⊢ ¬ 𝐴 = 𝐵

Syntax hints:  ¬ wn 3   = wceq 1287   ≠ wne 2251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-ne 2252

This theorem is referenced by:  2dom  6455  updjudhcoinrg  6693  exmidomni  6719  exmidfodomrlemr  6749  exmidfodomrlemrALT  6750  ine0  7793  inelr  7979  xrltnr  9159  pnfnlt  9166  xrlttri3  9176  nltpnft  9188  3lcm2e6woprm  10862  6lcm4e12  10863  peano3nninf  11253  nninfsellemqall  11263